Equations of Circles Review HW
Diagram
Term
Circle
A
C
B
Definition
Example
Radius
Diameter
r
Equation
of a
Circle
(h,k)
Give the center and radius. (Simplest radical form)
1.
(x – 2)2 + y2 = 25
2.
(x + 7)2 + (y – 3)2 = 63
Write the equation of the circle with the given center and radius.
3.
C(4, –5) ; r = 6
Graphs of Circles
x2 + y2 = r2
2
2
5. (x + 1) + (y – 3) = 16
4.
C(0, 6) ; r = 3 5
Center (0, 0) with radius r
6. (x – 2)2 + y2 = 20
Center:
Center:
Radius:
Radius:
Given the endpoints of the diameter of the circle write its equation.
7.
(7,8) and (-3,6)
Equations of Circles Review HW
8.
(2,5) and (-3,1)
Diagram
Term
Circle
A
C
B
Definition
Example
Radius
Diameter
r
Equation
of a
Circle
(h,k)
Give the center and radius. (Simplest radical form)
1.
(x – 2)2 + y2 = 25
2.
(x + 7)2 + (y – 3)2 = 63
Write the equation of the circle with the given center and radius.
3.
C(4, –5) ; r = 6
Graphs of Circles
x2 + y2 = r2
2
2
5. (x + 1) + (y – 3) = 16
4.
C(0, 6) ; r = 3 5
Center (0, 0) with radius r
6. (x – 2)2 + y2 = 20
Center:
Center:
Radius:
Radius:
Given the endpoints of the diameter of the circle write its equation.
7.
(7,8) and (-3,6)
8.
(2,5) and (-3,1)
3.1 Segments and Lines of Circles
Diagram
L
Term
Definition
Example
Chord
N
P
M
Secant
Tangent
R
O
Q
Point of Tangency
Theorem 3-1
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Theorem 3-2 (Converse of Theorem 3-1)
If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the
circle.
.
Corollary
Tangents to a circle from a point are congruent.

A line tangent to two circles in the same plane is called a common tangent.
o Common external tangents do not intersect the segment whose endpoints are the centers of the circles.
o Common internal tangents intersect the segment whose endpoints are the centers of the circles.
Example 1:
Determine whether RT is tangent to circle S. Explain.
R
16
12
P
S
T
8
Example 2:
TWR is circumscribed about circle A. If the perimeter of TWR is 42, MR = 6, and WM = 7, find TR.
T
L
K
R
W
M
Example 3:
LK and LE are tangent to circle M, mEML  66, KM  15, and LK  36. Find the following measures.
E
A. mMKL
M
P
L
K
B. mELM
C. EL
Name a line that
a. Tangent to circle P, but not to circle O.
b. Common external tangent to circles O & P.
c. Common internal tangent to circles O & P.
In the diagram below, circle M and circle N are tangent at P. PR and SR are tangents to circle N.
Circle N has diameter 16, PQ = 3, and RQ = 12.
Hint: For many problems, you will use right triangles.
a. PM =
R
Q
b. MQ =
c. PR =
d. SR =
e. NS =
f. NR =
P
M
N
S
3.2 Chord Relationships
Theorem 3-4
In the same circle or in congruent circles:
(1) Congruent arcs have congruent chords.
(2) Congruent chords have congruent arcs.
Theorem 3-5
A diameter that is perpendicular to a chord bisects
the chord and its arc.
Ex 1: Find x and y
y
x
Theorem 3-6
In the same circle or in congruent circles:
(1) Chords equally distant from the center are
congruent.
(2) Congruent chords are equally distant from the
center.
Ex 2: Find x, y, and m AB
Ex 3: Find x and y
60o
y
5
13
6
A
x
8
y
4
B
x
Theorem 3-7
If two chords intersect inside a circle, then the product of the segments of one chord is equal to the product
of the segments of the other chord.
Ex 4: Find x
Ex 5: Find x
Theorem 3-8
If two secants intersect outside a circle, then the product of the entire secant segment with its external portion
is equal to the product of the other entire secant segment with its external portion.
Ex 6: Find x
Ex 7: Find x
3.3 Arcs and Angles
Diagram
Term
Central
Angle
Inscribed
angle
Exterior
Angle
Arc
Definition
Example
Minor Arc
Major Arc
Definition of Arc Measure:
o The measure of a minor arc is the measure of its central angle.
o The measure of a major arc is 360o minus the measure of the central angle.
Ex 1:
̂ , 𝑚 𝐶𝐴𝐷
̂ , 𝑚 𝐴𝐷
̂ , and 𝑚 𝐷𝐶𝐴
̂.
In circle Q, AC is a diameter and mCQD = 40o. Find 𝑚 𝐶𝐷
̂ =
𝑚 𝐶𝐷
D
A
Q
̂=
𝑚 𝐶𝐴𝐷
C
̂ =
𝑚 𝐴𝐷
̂=
𝑚 𝐷𝐶𝐴
Find the measure of angle 1 in each diagram.
Ex 2:
Ex 3:
40o
Ex 4:
1
1
1
130o
30o
225o
Ex 5: Find the measure of each arc.
̂ =
a. 𝑚 𝐴𝐵
C
̂ =
b. 𝑚 𝐵𝐶
̂ =
c. 𝑚 𝐶𝐷
(2x-14)o
D
(4x)
(2x)
o
E
o
(3x)
B
o
̂ =
d. 𝑚 𝐷𝐸
̂ =
e. 𝑚 𝐸𝐴
A
(3x+10)o
Theorem 3-9
The measure of an inscribed angle is equal to half
the measure of its intercepted arc.
Corollary 1
If two inscribed angles intercept the same arc, then
the angles are congruent.
Theorem 3-10
The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of
the measures of the intercepted arcs.
Theorem 3-11
The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a
point outside a circle is equal to half the difference of the measure of the intercepted arcs.
Case I: Two secants
Case II: Two tangents
Case III: A secant and a tangent
60o
50o
xo
Example 6:
If m AB = 120, mBC = 30, and m CD = 160, find
mQ .
Q
B
C
A
D
Example 7: Find the value of x and m AET .
A
o
(40x - 2)
M
(100 + 4x)o
E
K
T(30x + 10)o
H
(183 - 5x)o
Example 8:
In circle K, m OB = 98, m OY = 28, m YD = 62, and m DA = 38. Find each measure.
B
1
K
O
2
m AB
3
Y
4
m 1
m 2
m 3
A
D
Example 9: Find the value of x
H
S
106o
xo 26o
Q
T
̂ =
Example 10: If mA  60 find 𝑚 𝐶𝐵
C
A
B
Example 11:
D
B
In circle O, AB is tangent, AF is a diameter, and
6
C
mAG = 100, mCE = 30, and mEF = 25.
E
Find the missing angles #1-8
8
O
3
A
2
5
1
7
G
4
F
Practice Problems – Find the x
1.
2.
#1 In circle P, AB  AC . Find the value of x to the nearest tenth.
B
4
6
P
x
R
A
C
#2 In circle R, TR = 6.4 and EN = 10.8. Find RO to the nearest tenth.
E
T
O
V
R
S
N
#3 In circle O, MO = 6 and LN = 16. Find x.
L
M
O
x
N
3.